Tensor Diagonalization Algorithms In Blind Source Separation

Tensor diagonalization,an important method in signal processing and machine learning,means transforming a given series of tensors to an exactly or nearly diagonal form through multiplying the tensor by nonunitary invertible matrices along selected dimensions of the tensor.For multi-dimensional,multi-dataset or multi-modal blind source separation,transforming the joint blind source separation problem to tensor diagonalization of high-order tensors by computing high-order cumulant of source in each dataset.And the applications of tensor diagonalization range from source separation to collaborative filtering,mixture and topic modeling,classification,and multilinear subspace learning.In this paper,we use tensor diagonalization as the entry point to study the algorithm of multi-dimensional,multi-dataset or multi-modal blind signal separation.The main contributions of this thesis are summarized below.We first study the complex value tensor diagonalization method in which the nonunitary diagonalizer may change with time.The existing tensor diagonalization algorithms consider that diagonalizer matrix remains unchanged.In practice,the number of target tensors may increase with time.We present an adaptive nonunitary tensor diagonalization algorithm to solve this situation.The algorithm recursively minimizes a least squares criterion and computes the diagonalizer by simply updating its previous estimate using tensor and vector calculations.Using recursive least squares criterion and tensor vectorization,the problem is transformed into a typical restricted least squares problem.We can solve this restricted least squares problem in two steps.Firstly,an unconstrained least squares problem related to the original problem is solved,and then the value of the optimal solution closest to the unconstrained least squares problem is found on the constraint set.When we compare this algorithm with other batch algorithms,we find that the algorithm has great advantages in carrying time.We have studied the multi-dataset nonunitary tensor diagonalization method.For the orthogonal or unitary diagonalizer problem,a pre-whitening process phase is required,which will reduce the performance of the algorithm.We study the nonunitary or nonorthogonal complex-valued tensor diagonalization problem with better algorithm performance.The algorithm is inspired by the idea of the Jacobi rotations framework.On this basis,each tensor mode is updated by combining a special non-zero parameter structure of the two positions (i,j) and (i,j).Corresponding different diagonalizer matrices until convergence.When we compare this algorithm with other batch algorithms,we find that the stability and diagonalization metric accuracy are superior when the algorithm is comparable to the computational complexity of other batch algorithms.